Strong altruism can evolve in randomly formed ... - Semantic Scholar
ARTICLE IN PRESS
Journal of Theoretical Biology 228 (2004) 303–313
Strong altruism can evolve in randomly formed groups Jeffrey A. Fletcher*, Martin Zwick Systems Science Ph.D. Program, Portland State University, Portland, OR 97207, USA Received 25 September 2003; accepted 21 January 2004
Abstract Although the conditions under which altruistic behaviors evolve continue to be vigorously debated, there is general agreement that altruistic traits involving an absolute cost to altruists (strong altruism) cannot evolve when populations are structured with randomly formed groups. This conclusion implies that the evolution of such traits depends upon special environmental conditions or additional organismic capabilities that enable altruists to interact with each other more than would be expected with random grouping. Here we show, using both analytic and simulation results, that the positive assortment necessary for strong altruism to evolve does not require these additional mechanisms, but merely that randomly formed groups exist for more than one generation. Conditions favoring the selection of altruists, which are absent when random groups initially form, can naturally arise even after a single generation within groups—and even as the proportion of altruists simultaneously decreases. The gains made by altruists in a second generation within groups can more than compensate for the losses suffered in the first and in this way altruism can ratchet up to high levels. This is true even if altruism is initially rare, migration between groups allowed, homogeneous altruist groups prohibited, population growth restricted, or kin selection precluded. Until now random group formation models have neglected the significance of multigenerational groups—even though such groups are a central feature of classic ‘‘haystack’’ models of the evolution of altruism. We also explore the important role that stochasticity (effectively absent in the original infinite models) plays in the evolution of altruism. The fact that strong altruism can increase when groups are periodically and randomly formed suggests that altruism may evolve more readily and in simpler organisms than is generally appreciated. r 2004 Elsevier Ltd. All rights reserved. Keywords: Altruism; Haystack model; Multilevel selection; Positive assortment; Randomly formed groups
1. Introduction Nearly three decades ago Hamilton (1975) and Wilson (1975) independently developed models which were interpreted as showing that strong altruism (involving an absolute cost to altruists) cannot evolve in randomly formed groups. This conclusion is still generally accepted even among those who debate how best to define altruism and the mechanisms by which it evolves (Hamilton, 1975; Maynard Smith, 1998; Nunney, 1985, 2000; Sober and Wilson, 2000; Wilson, 1975, 1990). Here we challenge this conclusion by exploring what happens when groups exist for more than one generation. Multigenerational groups are a central feature of Maynard Smith’s (1964) classic ‘‘haystack’’ *Corresponding author. Tel.: +1-503-725-4995; fax: +1-503-7258489. E-mail addresses: [email protected] (J.A. Fletcher), [email protected] (M. Zwick). 0022-5193/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2004.01.004
model, but the role of multiple generations within groups was not explored in Hamilton’s (1975) and Wilson’s (1975) models. Although the initial conditions after random group formation favor non-altruists over altruists, paradoxically these conditions can switch to favor altruists after even one generation of selection. Thus even though the overall proportion of altruists decreases after one generation, it can increase even more after a second generation spent within groups. Besides single-generation groups, these original analytic models rely on other simplifying assumptions such as an infinite population and no migration between groups. We begin by showing how strong altruism can evolve under the assumptions of the original models, with the only modification being delayed reformation of random groups. Multigenerational groups introduce additional issues such as interactions among related offspring, persistent homogeneous groups of altruists, and exponential growth of population size. We explore model modifications—preventing altruists from benefiting
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J.A. Fletcher, M. Zwick / Journal of Theoretical Biology 228 (2004) 303–313
kin, precluding homogeneous groups from forming, and adding a population-level carrying capacity—that mitigate each of these factors. We find that under all these modifications (imposed both separately and concurrently) strong altruism, although dampened, can still evolve in randomly formed multigenerational groups. We then transform the basic analytic model into an evolutionary simulation in which population size is finite and stable. In the simulation model random group formation from a finite population and migration between groups both reduce positive assortment and therefore dampen selection for altruism. On the other hand, increased stochasticity in benefit distribution and culling due to carrying capacity can enhance the likelihood that altruism will evolve compared to processes with minimum stochasticity.
2. Classifications of altruism Both the analytic and simulation models discussed here involve what Pepper (2000) has termed an otheronly altruistic trait because none of the altruist’s benefits come back to itself, as opposed to whole-group traits (also called group-beneficial traits) where the benefit is divided among all group members including the altruist. Wilson (1979, 1990) previously classified altruistic traits in a related but different way as either strong (involving an absolute cost to altruists) or weak (involving only a relative cost to altruists). Other-only altruistic traits are always strong while whole-group traits are strong if the cost to an altruist is greater than its share of the benefit it provides. Note that the same whole-group behavior involving the same sacrifice and provided benefit may be strong or weak depending on group size (Pepper, 2000). In contrast to strong altruism, Wilson (1979, 1990) showed that weakly altruistic traits can increase when groups are randomly formed every generation. That is, for an infinite population where a binomial trait is randomly redistributed every generation, the resulting between-group component of total variance can be enough for weak, but not for strong, altruism to evolve. Nevertheless in finite populations where fitness is relative the distinction between strong and weak altruism may be less important as both types are selected against within groups and require selection (or differential productivity) among groups in order to increase (Wilson, 1979, 1990). In this paper we focus on other-only, strong altruism (the most restrictive situation) to address the random group models of Hamilton and Wilson directly, but the consequences of multigenerational groups and stochasticity also apply to weak, whole-group traits and therefore these traits can even more readily increase via randomly formed groups than was previously shown (Wilson, 1979, 1990).
3. Analytic model We focus on Hamilton’s (1975) model because he developed a formal proof that altruism cannot evolve in single-generation randomly formed groups (Wilson’s (1975) model is similar in all important aspects). In this model a haploid infinite population is randomly subdivided into groups of equal size, n. Group members interact for one generation, affecting each other’s fitness (offspring count), before the population is pooled and then again randomly assigned to new groups. In every generation each altruist behaves in a way that costs itself c offspring and provides a total benefit of b offspring divided evenly among the other n-1 group members. Each non-altruist receives its share of benefits, but does not provide any benefit to others. Therefore, within every group non-altruists have more offspring than altruists, but groups with more altruists have more offspring per capita than groups with less. This is an example of multilevel selection where here selection within groups opposes selection between groups. Hamilton (1975) using Price’s (1970) covariance equation showed that under his model’s assumptions, between-group selection (due to the variance between groups in altruist frequency, p) must always be weaker than average within-group selection (due to the expected variance in the altruistic trait within groups) and therefore the overall frequency of altruists, P, must decrease in every generation. (Capital letters indicate whole population values; small letters indicate group values.) To illustrate this, we calculate D1 P for an infinite binomial distribution, where D1 indicates that the change occurs over one generation within groups, g ¼ 1. The variable g is the number of generations spent within groups before each reformation event. (See Appendix A for model details.) Fig. 1(a) shows how D1 P depends on the level of benefit, b, provided by altruists for different starting P values. (For convenience, all results reported in this paper use c ¼ 1 such that benefit b is also the benefit to cost ratio.) The results shown in Fig. 1(a) are the same for any group size n. Note that as benefit increases, zero is an upper limit on D1 P—hence the conclusion that strong altruism cannot increase under the assumptions of this model for all values of P and n (Hamilton, 1975; Wilson, 1975). Yet quite different results are obtained if groups persist for even one additional generation ðg ¼ 2Þ before random mixing and the formation of new groups. Fig. 1(b) shows how the change in P after two generations within groups, D2 P; depends on benefit values for different starting P values. The only difference between Figs. 1(a) and (b) is that the latter measures the change in altruist frequency after an additional generation spent within groups where the fitness functions are deterministically and recursively
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for an altruistic trait to increase in the next generation is
0.05
rb > c;
∆1P
0 -0.05
P = 0.9 P = 0.75 P = 0.25 P = 0.1
-0.1 -0.15 0
20
40
60
80
100
b
(a) 0.3
∆2P
0.2 0.1 0 P = 0.1 P = 0.25
-0.1 0
20
40
(b)
b
60
P = 0.75 P = 0.9 80
100
0.008 n=2 n=3 n=4 n=5
0.006
∆2P
305
0.004 0.002
where r is the ‘‘coefficient of relatedness’’ or more generally the regression coefficient between the frequency of the trait in recipients and actors (Hamilton, 1972). Thus r is a measure of positive assortment—the degree to which the benefits of altruists fall to other altruists. The value of r differs for other-only and wholegroup traits because in the latter case, but not the former, altruists are recipients of their own actions (Pepper, 2000). We use superscripts w and o, respectively, for whole-group and other-only measures of r. For whole-group traits rw is the between-group variance in p over the total variance in the altruistic trait (Breden, 1990; Frank, 1995a). For an infinite binomial population of randomly formed groups of size n, the variance ratio rw ¼ 1=n: Thus according to Hamilton’s rule (Eq. (1)) the trait increases after one generation if b=n > c; but for whole-group traits this means that an altruist’s share of its benefit must be greater than its cost—this is the definition of weak altruism so as Wilson (1979, 1990) noted only weak traits can increase after one generation. For groups of uniform size the r values are related by the following expression (Pepper, 2000):
0
ro ¼
-0.002 0
20
40
(c)
60
80
100
b
Fig. 1. Change in altruist frequency ðDg PÞ as a function of altruist benefit, b. (a) and (b) compare the effect of different starting P values after one ðg ¼ 1Þ and two ðg ¼ 2Þ generations spent within groups, respectively, where founding group size n ¼ 4 (although in (a) the results are the same for all n); (c) compares the effect of different n for multigenerational groups ðg ¼ 2Þ when altruism is rare—here P ¼ 0:001: The cost c ¼ 1 in all calculations.
applied. In this case strong altruism can clearly increase ðD2 P > 0Þ for sufficient values of benefit. Fig. 1(c) shows that smaller groups give a larger increase in altruist frequency which is consistent with previous findings on the relationship between group size and the evolution of altruistic traits (Avile! s, 1993; Boyd and Richerson, 1988). Additionally Fig. 1(c), for which P = 0.001, shows that strong altruism can increase due to multigenerational groups even when the altruistic trait is rare, although higher benefit levels are needed for D2 P > 0 when P is low.
4. Applying Hamilton’s rule We can also understand these results in terms of Hamilton’s rule (1964) which states that the condition
ð1Þ
nrw 1 : n1
ð2Þ
Therefore ro ¼ 0 for an initial random distribution where rw ¼ 1=n: Obviously there are no positive values of b and c that can satisfy Hamilton’s rule (Eq. (1)) for an other-only (strong) altruistic trait when r ¼ 0 and such traits must decrease. Note however, that any modifications to the model that make ro > 0 can yield an increase in P, given a sufficient value of b. Hamilton (1975) noted that any positive assortment of altruists beyond that produced at random could allow altruism to increase. Surprisingly, for many parameter settings ro increases above zero after one generation of selection— even as the proportion of altruists decreases. That is, this transient one-generation-long ‘‘population viscosity’’ of the original models is enough (without any other mechanisms for creating positive assortment) to create conditions that favor altruism in the following generation. If groups are randomly reformed after this single generation then this gain in positive assortment is destroyed before being used by selection; ro returns to zero and P subsequently declines. On the other hand, additional generations within groups can take advantage of this increased positive assortment so that strong altruism increases, as shown in Fig. 1. (Whether altruism actually increases or not depends on parameters including P, b, and n.) Note that although Hamilton emphasized a ratio of variances in his proof, in this other-only model the
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306
regression coefficient between actors and recipients, ro, is an easier to interpret measure of the changing conditions affecting altruism as selection occurs. For instance, after one generation of selection (starting with randomly formed groups), the between over total variance, rw, can decrease while ro increases. It is the increase in ro that accurately reflects whether altruism can increase in the next generation. For most parameter settings both measures increase after one generation, but the range of parameters where rw decreases is greater than the range in which ro decreases. In general the r values can decrease when there is a combination of low P, low b, and high n. According to Hamilton’s rule whether altruism increases in the second generation within groups depends on whether ro after the first generation is greater than c/b. Of course, it is not enough for altruism to increase in the second generation for altruism to increase overall. The increases in subsequent generations within groups must make up for any losses in the initial generation(s). Fig. 2(a) shows the expected dynamic change in P values over successive generations when groups persist for one and two generations before
random reformation and Fig. 2(b) shows the concurrent changes in ro. Altruist frequency P decreases monotonically when groups are reformed every generation and ro ¼ 0 before each round of selection. On the other hand when groups exist for two generations, P oscillates (and can ratchet upward). The every-other-generation saw-toothed peaks in P correspond to similar (but offset) oscillating peaks in ro (Fig. 2(b)). Here ro increases after a generation within groups and we indicate the critical c/b value with a solid horizontal line. Troughs on the other hand correspond to global mixing, new group formation, a decrease of ro back to zero and a subsequent decrease in P. In Fig. 2 we also show a case with the same parameters except bigger group size (g ¼ 2; n ¼ 10). Here, although P can increases during the second generations within groups, it is not enough to make up for losses in the first generations. Note that when peaks in ro fail to reach the c/b value (after generation 21 in Fig. 2(b) for g ¼ 2; n ¼ 10), as predicted by Hamilton’s rule, P can no longer increase and instead falls during both generations within groups (Fig. 2(a)).
5. Analytic model modifications 1
g = 1; n = 3 g = 2; n = 10 g = 2; n = 3
P
0.75
0.5
0.25
0 0
5
(a)
10 15 20 total generations
25
30
0.5
ro
0.25 c/b
0
-0.25 0
(b)
5
10 15 20 total generations
25
30
Fig. 2. Calculated dynamics in the analytic model for one and two generations spent within groups ðg ¼ 1; 2Þ for groups of two different sizes (n ¼ 3; n ¼ 10); (a) shows the dynamics in overall altruist frequency P given different starting P values and different sized groups; (b) shows the concurrent change in the regression coefficient between actors and recipients, ro. The critical ro value of c/b is also shown with a solid horizontal line. Here P ¼ 0:1; b ¼ 10; and c ¼ 1: Both ro and P are calculated at the end of the indicated generation and after group reformation if it occurs; (a) and (b) use the same legend.
Multiple generations within groups complicate the simple single-generation model in several ways: (1) kin interactions within groups become possible: (2) the contribution of homogeneous groups of altruists increases—these groups uniquely retain their initial (maximal) level of altruism; (3) the additive frequencydependent fitness functions can now lead to exponential growth of the population. Yet as we demonstrate below, while not inconsequential, none of these factors are essential to explain why strong altruism increases in randomly formed multigenerational groups—especially when altruism is initially rare. The following three paragraphs elaborate on each issue and describe modifications to the basic model to address them. We follow this with a summary of the results produced by each modification. 5.1. No kin selection In the original model groups are formed by randomly selecting individuals from an infinite population and therefore groups contain unrelated individuals. In a second generation within groups, when the benefits provided by an altruist are divided among other group members, some of this benefit (in the form of additional offspring) will fall to those with the same parent as the altruist. In general the proportion of benefit falling to relatives (defined by common ancestry) in subsequent generations will depend on parameters n, b, and P, but this proportion is bounded by 1/n (Appendix B). This
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5.2. No homogeneous groups
0.2
0.1
∆gP
limit is approached for high b and P, but when altruism is initially rare this proportion is much less significant. For example, for groups of size 4, the maximum possible proportion of benefit that could fall to relatives is 1/n=25%, but for P ¼ 0:1 and b ¼ 10 we observe the proportion of benefit given to relatives is actually 0.61% during the second generation within groups. For the same P and b the proportion decreases with larger group size and more generations within groups. To eliminate kin selection we modify our model so that altruists only divide their benefit among non-relatives (Appendix B).
0 g = 2; unmodified g = 2; no homogeneous groups g = 2; no populaiton growth g = 2; no kin benefit g = 2; all 3 modifications g = 1; unmodified
-0.1
-0.2 0
In the infinite population of this model, homogeneous groups of altruists will be randomly created whenever P > 0: These groups are unique in being the only group composition for which p does not decrease with successive generations within groups. They are also the fastest growing groups as they contain no free-riding non-altruists. One might suspect that such homogeneous groups account for altruism being able to increase after multiple generations within groups. To check this we modify our model (Appendix B) such that immediately after group formation all homogeneous groups of altruists have one altruist switched to a non-altruist. Note that this artificially decreases P, making it even harder for altruism to evolve. 5.3. No population growth Even with additive (linear) fitness functions, multiple generations within groups can cause a population to grow exponentially (Wilson, 1987). To study the effect of stable population size we implement a global carrying capacity by scaling the offspring count of all population members each generation by the inverse of the expected overall growth rate during that generation (Appendix B). This holds the population size constant (albeit infinite) at every generation, but allows groups with more altruists to have relatively more offspring each generation than groups with less. 5.4. Modification results Fig. 3 compares the results for each of these modifications with the unmodified model for two generations within groups, g ¼ 2: We also include results for the original model where g ¼ 1: For each of the three modifications D2 P is dampened, but still positive given sufficient benefit. This is true even when all of the modifications are imposed simultaneously. That is, in a model where no benefit is given to kin, homogeneous groups are always corrupted, and population size is held constant, strong altruism can still increase after two generations within groups.
307
20
40
60
80
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b
Fig. 3. Change in altruist frequency ðDg PÞ after two ðg ¼ 2Þ generations within groups as a function of altruist benefit, b, for several modifications of the original binomial model including preventing homogeneous groups from forming, scaling the population size to its original size each generation, and distributing altruist benefit only to the non-relatives of an altruist. The results for the original model after one ðg ¼ 1Þ and two ðg ¼ 2Þ generations are also shown for comparison. The original P ¼ 0:25; n ¼ 4; and c ¼ 1 in all calculations.
In contrast to the unmodified dynamic model shown in Fig. 2 where altruism tends to evolve to P ¼ 1:0 or 0:0 given enough generations, it does not necessarily evolve to saturation under all these modifications. Corrupting homogeneous groups for example necessarily keeps Po1:0: In the case of a population-level carrying capacity, for n ¼ 4; P ¼ 0:1; g ¼ 2; and b ¼ 15; a stable limit cycle is reached in which P oscillates every other generation between 0.616 and 0.636. (Yet as shown in the next section, when stochasticity is introduced populations tend to evolve to one extreme or the other in these models.)
6. Simulation model So far, like Hamilton, we have used the assumption of an infinite population in order to calculate the expected distribution of group compositions when individuals are randomly distributed. But infinity here has two special consequences. First it converts a seemingly stochastic process (random group formation) to a deterministic one—the expected value of ro is produced by every group reformation event. For any finite population, group reformation events will produce ro values that fluctuate both above and below the average value. Second, the expected value of ro is lower (i.e. less than zero) for groups formed randomly from a finite population (compared to an infinite one). This is because in the finite case where ‘‘sampling without replacement’’ is used we have a hypergeometric (rather than binomial) distribution. Once an individual of a certain type is
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assigned to a random group, the probability of assigning another individual of that type is lower than in the first assignment. This results in the formation of less homogeneous groups and more heterogeneous groups (than in the infinite binomial distribution) which decreases the overall assortment—thus on average ro is negative. Based on empirical calculations it appears that the expected negative value of ro depends only on population size (not n or P) and is ro ¼ 1=ðN 1Þ for equal-sized groups randomly formed from a finite population of size N. Particular random group formation events will result in ro values which fluctuate around this lower value. In addition, a finite simulation model can incorporate the consequences of other stochastic processes including the way altruistic benefit is distributed and the way individuals subject to a carrying capacity are eliminated. We now transform the analytic model above into a computer simulation of a finite evolving population and the following features and options (see Appendix C for further details): 1. When reforming groups each individual is assigned at random to an unfilled group (rather than by using the expected distribution). 2. The benefit value used by each group, each generation and the distribution of this benefit to other group members is done in one of two ways: (i) Low stochasticity: the benefit level is the same for all groups during a simulation run and is divided as evenly as possible (in units of whole offspring) with only any remainder distributed randomly among other group members. (ii) High stochasticity: the benefit level used in a group is drawn from a Poisson distribution in whole units where the given benefit value is the mean and each unit of benefit is then distributed at random to other group members. 3. Population size is held constant each generation by a global carrying capacity in one of two ways: (i) Low stochasticity: group sizes are proportionally scaled back (as in the no-growth analytic model), but only whole organisms are removed proportionally with any remainder removed randomly (as below). (ii) High stochasticity: excess population offspring are removed at random (without regard to the altruistic trait or group membership). The first options in 2 and 3 above minimize stochasticity while still preserving the simulation model’s integral organisms, whereas the second options introduce more stochasticity. Fig. 4 compares the change in altruist frequency after 2 generations within
0.2
∆2P
308
0.1
both stochasic options added stochastic elimination added analytic model, no growth stochastic benefit added stochastic group formation
0 0
10
20
30
40
50
b
Fig. 4. Change in altruist frequency after two generations within groups ðD2 PÞ as a function of altruist benefit, b, for several variations of the finite simulation model with no population growth. The base case is stochasticity in random group formation, but otherwise minimal noise. To this case we add high stochasticity in benefit distribution, high stochasticity in implementing carrying capacity, and both options simultaneously. Each data point is the average of 1000 runs done with population size N ¼ 1000: The other parameters for all runs are P ¼ 0:25; n ¼ 4; and c ¼ 1: The results for the infinite analytic model with no population growth using the same parameters are also shown for comparison.
groups, D2 P; for the no-population-growth run of the analytic model (from Fig. 3), the simulation model with stochastic group formation but minimal other stochasticity, and each of the more stochastic choices introduced separately, and then simultaneously. Each data point represents the average of 1000 runs done with different random number seeds. Note that strong altruism evolves less easily in the finite simulation with stochastic group formation (than in the comparable infinite analytic model). From this base, adding more stochasticity in benefit distribution has a slight positive effect whereas adding more stochasticity in elimination due to carrying capacity has a strong positive effect. The latter case results in an even bigger increase in altruism than in the no population growth analytic model. Using both options simultaneously does even better. Our methods of adding stochasticity are somewhat ad hoc and we do not imply that additional stochasticity will necessarily increase D2 P: In fact, the stochasticity in random group formation appears to dampen selection for altruism—D2 P is less than would be expected in an average hypergeometric distribution. The effect of a particular method of introducing random noise will depend on its relative impact on within- and betweengroup selection. We do however show that it is possible for altruism to evolve even more easily in stochastic finite populations than it does in deterministic infinite models. We now explore the long-term behavior of our simulation. Because here there is no mutation, empirically
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% runs P = 1.0
100 80 60
P = 0.90 P = 0.25 P = 0.10 P = 0.05
40 20 0 0
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10 b
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100 n=2 n=4 n=6 n=8
80 60 40 20 0 0
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Fig. 5. The percentage of simulation runs reaching altruist saturation as a function of altruist benefit, b; (a) compares the effect of different starting P values, where the number of generations spent within groups g ¼ 2 and initial group size n ¼ 4; (b) compares the effect of different n where P ¼ 0:1 and g ¼ 2: The total population size N ¼ 1000 and cost c ¼ 1 in all runs. 1000 runs were done for each unique set of parameters with different random number seeds. All runs were done until P ¼ 1:0 or 0.0. Here the high stochasticity carrying capacity option and the low stochasticity benefit distribution are used.
m = 0.0
m = 0.1
m = 0.2
m = 0.3
100
% runs P = 1.0
we observe that P ¼ 1:0 and 0.0 act as stable equilibrium points and intermediate values do not persist indefinitely. All runs were done until one of these equilibrium points was reached and we use the percentage of 1000 runs reaching altruist saturation, P ¼ 1:0; as a measure of how readily altruism evolves under the given conditions. For all runs and figures described in the rest of this paper we use the high stochasticity option in implementing carrying capacity, but minimum stochasticity in benefit distribution. If the results shown in Fig. 4 apply generally, then these settings are more favorable to the evolution of altruism than the comparable analytic model, but less favorable than if we had used high stochasticity in both processes simultaneously. As was the case for the analytic model, Fig. 5(a) shows that both higher starting P and higher benefit values favor selection for altruism and Fig. 5(b) shows that less altruistic benefit is required to evolve altruism for smaller group sizes. We now investigate the effect of migration in our simulation model where the migration rate, m, specifies the probability that an individual will leave its group during each generation, moving to a randomly selected group (weighted proportionately by group size). The idea here is that larger, thriving groups are proportionately more attractive to migrants, but similar results obtain when migrants join groups at random, independent of group size. Fig. 6 shows how the interaction between the number of generations spent within groups
309
80 60 40 20 0 0
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4
g
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10
Fig. 6. The percentage of simulation runs reaching altruist saturation as a function of the number of generations spent within groups g. Compares the effect of different migration rates, m, where for all runs P ¼ 0:1; g ¼ 2; b ¼ 10; n ¼ 4; N ¼ 1000; and c ¼ 1: 1000 runs were done for each unique set of parameters with different random number seeds. All runs were done until P ¼ 1:0 or 0. For m ¼ 0:4 all runs resulted in P ¼ 0:0 (data not shown). High stochasticity carrying capacity and low stochasticity benefit distribution are used.
and the migration rate influence selection for strong altruism. Predictably, migration lessens selection for altruism (Fig. 6) by working to dampen the positive assortment, ro, between actors and recipients each generation, but for intermediate numbers of generations spent within groups, even at relatively high migration rates (i.e. 30%), strong altruism evolves to saturation in some runs. Fig. 6 also shows that even without migration ðm ¼ 0:0Þ intermediate numbers of generations within groups are most favorable to the evolution of altruism. The advantage of an intermediate number of generations is consistent with similar findings in haystack models (Wilson, 1987) and models of biased sex ratios (Wilson and Colwell, 1981). Note that in the simulations of Fig. 6 it is initially unlikely that any homogeneous groups of altruists will form. With initial P ¼ 0:1 and n ¼ 4 the probability of forming homogeneous altruist groups is one in 10,000 and only 250 groups are formed ðN ¼ 1000; n ¼ 4Þ at each group reformation. Yet, in the absence of homogeneous groups strong altruism can still initially increase overall even as p declines in every group. This is because groups with a higher frequency of altruists grow faster—population P increasing while every group p decreases is an example of Simpson’s paradox (Simpson, 1951; Sober and Wilson, 1998). With more generations within groups P must eventually decrease as the altruists are eliminated from every group. Altruism evolves most readily when the number of generations spent within groups takes full advantage of the increase in P due to Simpson’s paradox, but avoids the inevitable decline in P. Fig. 7 illustrates this tension. Here representative individual runs are shown for 2, 4, and 10 generations within groups using the same parameters as Fig. 6 without migration ðm ¼ 0:0Þ: To aid in comparison the same random number seed (same initial group distribution) is used in all three runs. For 10 generations within groups ðg ¼ 10Þ; reformation clearly takes place well
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310 1 0.8
P
0.6 g=2 g=4 g = 10
0.4 0.2 0 0
5
10 15 20 total generations
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Fig. 7. Dynamics in altruist frequency P for individual simulation runs of g ¼ 2; 4; and 10 generations spent within groups. For all runs initial P ¼ 0:1; b=10, n ¼ 4; N ¼ 1000; and c ¼ 1: To aid in comparison, all three runs were initiated with the same random number seed (same initial distribution into groups). High stochasticity carrying capacity and low stochasticity benefit distribution are used.
after peak P values are reached and altruism eventually goes extinct. For g ¼ 4; reformation takes place near peak P values and altruism rapidly evolves towards saturation—even though altruism always decreases in the first generation after reformation. (This is true for g ¼ 10 also, but harder to see as the rate of decline after reformation matches the rate before.) On the other hand when groups are reformed every other generation ðg ¼ 2Þ; the potential additional increase in P that would result from staying within groups longer is lost and altruism increases more gradually. Note that the initial increase in P in these three runs takes place in the absence of homogeneous groups. For g ¼ 2 and 4 no such groups are formed until P reaches about 0.3 (initial P ¼ 0:1) and in the case of g ¼ 10 homogeneous groups never formed. In contrast, in the infinite analytic model homogeneous groups are always initially present and more generations within groups can allow these fastestgrowing groups to become more and more dominant, even if initially rare.
7. Conclusion The main purpose of this paper is to demonstrate that strong altruism can evolve in randomly formed groups and thereby challenge a presumed theoretic limitation on the evolution of altruism. Although allowing groups to last more than one generation introduces new complications, we have demonstrated that kin selection, homogeneous groups, and population expansion are not essential to account for this phenomenon. The fundamental explanation is that, for many initial conditions, after even just one generation of selection in randomly formed groups, the positive assortment between altruists and their potential recipients increases (above the expected initial value for randomly formed groups) as measured by the regression coefficient, ro. The groups
that are by chance initially dominated by altruists grow larger compared to other groups and even though the fraction of altruists declines in these groups, the absolute number of altruists poised to benefit other altruists in a subsequent generation increases. On the other hand, the groups that are by chance dominated by non-altruists do not grow as large and the relatively few altruists in these groups are eliminated or greatly diminished after one to several generations within groups. This also increases positive assortment as these non-altruists are stuck with each other and will receive less benefit from altruists than they did in the first generation. Of course the few non-altruists lucky enough to end up in altruistdominated groups are the fittest individuals, but overall the conditions that favored non-altruists in the initial random distribution can switch to favor altruists in subsequent generations. We emphasize again that even when groups are multigenerational, the vast majority of the benefit provided by altruists will fall to non-relatives—especially when altruism is initially rare (Appendix B). Altruism evolves due to the positive assortment among heritable helping behaviors regardless of whether there is a positive assortment among relatives by descent. The regression coefficient used here, ro, measures the former. This positive assortment can be viewed equivalently (Frank, 1998; Queller, 1985, 1992; Sober and Wilson, 1998; Wade, 1980) as causing selection on the altruistic trait (allele) via inclusive fitness or as causing selection among groups that vary in their trait composition. While interactions among kin in nature no doubt often contribute to the positive assortment of altruistic traits, kin interactions are not in themselves a requirement for altruism to evolve. Whether strong altruism evolves in nature via mechanisms similar to those illustrated here will depend on the degree to which the assumptions of these models are representative of natural conditions. For instance, in both the analytic and simulation models we demonstrated that strong altruism can evolve even when population size is held constant by a global carrying capacity. In nature, in addition to population-level limits on growth there are often limits on group size. While not explored here, group-level limits will dampen between-group selection for altruism, so further investigation is needed to elucidate the relative import of global vs. local levels of population control in the evolution of altruism. A lack of mutation is also unrealistic. We experimented with mutation in our models (data not shown), but in the simple binary genetics used here a mutation that switches behavioral types exerts pressure towards P ¼ 0:5 and thus favors altruism when P is initially low. This is because the more common type experiences more mutations. Even if this bias could be compensated for, low mutation rates are unlikely to alter our basic results, which are robust
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under fairly high levels of migration among groups and even when homogeneous groups of altruists are ‘‘mutated’’ to contain at least one non-altruist. This model started with the original assumptions of the random group models (Hamilton, 1975; Wilson, 1975) and added the idea of multigenerational groups from haystack models (Maynard Smith, 1964; Wilson, 1987). Just as Wilson (1987) created a simulation model to study a more realistic version of Maynard Smith’s (1964) original haystack model, we have created a simulation model that adds finite population size, stochasticity, and mutigenerational groups to the original analytic random group models. Whereas Wilson’s (1987) haystack simulation corrected the ‘‘worst case’’ assumption made by Maynard Smith (1964) that groups would persist until altruism was eliminated in all mixed groups; here we correct an opposite ‘‘worst case’’ assumption made in random group models that groups only exist for a single generation. As demonstrated here and in the haystack simulations (Wilson, 1987), an intermediate number of generations within groups is most favorable to the evolution of altruism. Maynard Smith (1998) in discussing different views on the evolution of altruism recently echoed the original findings of Hamilton (1975) and Wilson (1975) and the current consensus opinion when he wrote: ‘‘If costs and benefits combine additively, and groups are formed randomly, then altruism cannot evolve. But if altruists tend to associate with altruists, and non-altruists with non-altruists, then altruism can evolve. This conclusion is agreed.’’ Many mechanisms which result in a positive assortment among self-sacrificing behaviors have been proposed including passive methods such as foraging in non-uniform resource distributions which can be depleted (Pepper and Smuts, 2002), continuous population viscosity with periodic environmental disturbances (Mitteldorf and Wilson, 2000), the coevolution of group joining and cooperative behaviors (Avile! s, 2002), and the presence of non-participants (Hauert et al., 2002), as well as active methods such as kin recognition (Gamboa et al., 1991), conditional strategies based on past actions (Axelrod and Hamilton, 1981; Trivers, 1971) or reputation (Nowak and Sigmund, 1998; Panchanathan and Boyd, 2003), policing (Frank, 1995b, 2003), punishment of non-altruists (Boyd et al., 2003; Boyd and Richerson, 1992; Fehr and G.achter, 2002), the coevolution of cultural institutions that constrain individual behaviors (Bowles et al., 2003), and even recognition of arbitrary tags (Riolo et al., 2001). Here we have shown in both deterministic and stochastic models that when groups exist for more than one generation such specific or more complex mechanisms for creating positive assortment, although certainly important if present, are not needed—the positive assortment that develops between randomly created multigenerational groups can suffice
311
for between-group selection to dominate within-group selection and thus for strong altruism to evolve.
Acknowledgements We wish to thank Leticia Avile! s and Mark Bedau for helpful comments on an earlier draft of this manuscript and two anonymous reviewers for detailed and useful suggestions. We also thank Gene Enneking, Len Nunney, and Tad Shannon for helpful discussions.
Appendix A. Analytic model Here we describe Hamilton’s original model with recursion added to accommodate multiple generations within groups. If ag, sg, and ng are, respectively, the number of altruists, non-altruists (selfish individuals), and total individuals in a group after g generations spent within groups, then: ag1 1 ag ¼ ag1 1 þ b c ; ðA:1Þ ng1 1 ag1 sg ¼ sg1 1 þ b and ðA:2Þ ng1 1 ag1 ng ¼ ng1 1 þ ðb cÞ : ðA:3Þ ng1 In Hamilton’s model g is always one, but in our model we vary g by using these equations recursively— inputting the results from one generation into the calculations for the next. Note that when first formed all groups are size n, but after reproduction group sizes vary. (Terms without g subscripts indicate initial values, i.e. n is n0.) The overall number of altruists, Ag ; and individuals, Ng ; in the population after g generation within groups is then the number contributed (after g generations) by groups of every possible original composition (a = 0 to n) times the number of such groups expected in a random binomial distribution. If G is the total number of groups, then the expected count of groups with a initial altruists out of n group members is ! n hðaÞ ¼ G Pa ð1 PÞðnaÞ : ðA:4Þ a The total population values after g generations spent within groups are then given by n X Ag ¼ hðiÞ ag ðiÞ and ðA:5Þ i¼0
Ng ¼
n X i¼0
hðiÞ ng ðiÞ;
ðA:6Þ
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312
where ag(i) is the ag value from Eq. (A.1) given the particular starting a value specified by the index i and similarly for ng(i). Although G is infinite, it cancels in the calculation of Pg ¼ Ag =Ng and Dg P ¼ Pg 2P.
Appendix B. Analytic model modifications Here we describe modifications to the analytic model such that altruists do not give benefit to kin, homogeneous groups are disallowed, and the population size is kept constant.
B.2. No homogeneous groups In an infinite population there will always be some homogeneous groups whenever P > 0: For this modification, each time groups are randomly formed we eliminate homogeneous groups by simply converting all groups where the altruist number a ¼ n to groups where a ¼ n21: In this way all homogeneous groups of altruists have one member switched to a non-altruist. Because non-altruists always increase faster than altruists, groups become more non-homogeneous with each successive generation. Note that this modification causes P to decrease, but this effect is small when P is small.
B.1. No benefit to kin The modified fitness functions for when altruists only give to non-relatives are implemented by substituting the size of altruist kin groups, k, for the minus-one term in Eqs. (A.1) and (A.2). The minus-one term subtracted the altruist from the number of its beneficiaries; here we subtract the altruist’s kin (those having a common ancestor) as well. A preceding superscript k is used to designate fitness calculations that subtract k instead of one from a group’s altruist count and group size: k ag1 kg1 k k ag ¼ ag1 1 þ b c ; ðB:1Þ ng1 kg1 k
sg ¼ k sg1 1 þ b
ag1 ; ng1 kg1 k
ðB:2Þ
where the size of a kin group of altruists in generation g is given by ag kg ¼ kg1 : ðB:3Þ ag1 This is the size of the kin group in the last generation times an altruist’s clutch size for this generation. The initial k value k0 ¼ 1 (altruists are only related to themselves). Shifting benefit from kin to non-kin in this way does not affect the total group size and Eq. (A.3) works for calculating ng. Note that in the unmodified model the average proportion of a group that is related to an altruist, kg/ng, can never be above 1/n and therefore the proportion of an altruist’s benefit that falls to kin (kg–1)/(ng–1) is also bounded by 1/n. To see this note that kg/ng will be largest within homogeneous groups of altruists compared to mixed groups. In such groups (given our convention that c ¼ 1) kg is multiplied by b each generation and total group size also increases with b. Therefore the proportion kg/ng remains at its original value of 1/n. In all other groups this proportion falls with successive generations. Only when P is high (so that homogeneous altruist groups are common) or when b is high (so homogeneous altruist groups grow proportionally bigger than other groups) is this limit approached.
B.3. No population growth The global carrying capacity is implemented by scaling back all offspring numbers each generation by Ng1 =Ng where Ng is first calculated without scaling. We use a preceding asterisk to denote values calculated with scaling. For instance, the number of altruists in a group after g generations with scaling is, a ¼ a Ng1 g g Ng
ðB:4Þ
and similarly for group size ng, where scaling is imposed at each recursion (generation). Whole population values with scaling Ag and Ng then sum over ag and ng instead of ag and ng, respectively, in Eqs. (A.5) and (A.6) and Pg=Ag/Ng.
Appendix C. Computer simulation model For each run of the model, individuals ðN ¼ 1000Þ are initially randomly distributed into groups of size n using a random number generator to assign individuals to unfilled groups. The proportion of altruists and nonaltruists is determined by the starting P value. The sequential steps of the simulation are then: 1. In each group the new number of altruists and nonaltruists (to the closest whole individual) are determined (using either the low or high stochasticity method described in the text). 2. Individuals are eliminated (using either the low or high stochasticity method described in the text) until the original population size N is reached. 3. If g generations have passed within groups since the last group reformation, all individuals are randomly assigned to new groups of size n; otherwise if the migration rate, m, is greater than zero, mN individuals are chosen at random from the whole population and moved to new random locations in the population array which is ordered by
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groups—consequently larger groups are proportionately more likely to send out and receive migrants. These steps are repeated until an equilibrium at P ¼ 0:0 or 1.0 is reached. For runs where n ¼ 6; N was 1002 instead of 1000 and initial P ¼ 0:0998 instead of 0.1000 to allow an even distribution into groups.
References Avil!es, L., 1993. Interdemic selection and the sex ratio: a social spider perspective. Am. Nat. 142, 320–345. Avil!es, L., 2002. Solving the freeloaders paradox: genetic associations and frequency-dependent selection in the evolution of cooperation among nonrelatives. Proc. Natl Acad. Sci. 99, 14268–14273. Axelrod, R., Hamilton, W.D., 1981. The evolution of cooperation. Science 211, 1390–1396. Bowles, S., Choi, J.-K., Hopfensitz, A., 2003. The co-evolution of individual behaviors and social institutions. J. Theor. Biol. 223, 135–147. Boyd, R., Richerson, P.J., 1988. The evolution of reciprocity in sizeable groups. J. Theor. Biol. 132, 337–356. Boyd, R., Richerson, P.J., 1992. Punishment allows the evolution of cooperation (or anything else) in sizable groups. Ethol. Sociobiol. 13, 171–195. Boyd, R., Gintis, H., Bowles, S., Richerson, P.J., 2003. The evolution of altruistic punishment. Proc. Natl Acad. Sci. 100, 3531–3535. Breden, F., 1990. Partitioning of covariance as a method for studying kin selection. Trends Ecol. Evol. 5, 224–228. Fehr, E., G.achter, S., 2002. Altruistic punishment in humans. Nature 415, 137–140. Frank, S.A., 1995a. George price’s contributions to evolutionary genetics. J. Theor. Biol. 175, 373–388. Frank, S.A., 1995b. Mutual policing and repression of competition in the evolution of cooperative groups. Nature 377, 520–522. Frank, S.A., 1998. Foundations of Social Evolution. Princeton University Press, Princeton. Frank, S.A., 2003. Perspective: repression of competition and the evolution of cooperation. Evolution 57, 693–705. Gamboa, G.J., Reeve, H.K., Holmes, W.G., 1991. Conceptual issues and methodology in kin-recognition research: a critical discussion. Ethology 88, 109–127. Hamilton, W.D., 1964. The genetical evolution of social behavior i and ii. J. Theor. Biol. 7, 1–52. Hamilton, W.D., 1972. Altruism and related phenomena, mainly in social insects. Annu. Rev. Ecol. Syst. 3, 193–232. Hamilton, W.D., 1975. Innate social aptitudes of man: an approach from evolutionary genetics. In: Fox, R. (Ed.), Biosocial Anthropology. Wiley, New York, pp. 133–155.
313
Hauert, C., De Monte, S., Hofbauer, J., Sigmund, K., 2002. Volunteering as red queen mechanism for cooperators in public goods games. Science 296, 1129–1132. Maynard Smith, J., 1964. Groups selection and kin selection. Nature 201, 1145–1147. Maynard Smith, J., 1998. The origin of altruism. Nature 393, 639–640. Mitteldorf, J., Wilson, D.S., 2000. Population viscosity and the evolution of altruism. J. Theor. Biol. 204, 481–496. Nowak, M.A., Sigmund, K., 1998. Evolution of indirect reciprocity by image scoring. Nature 393, 573–577. Nunney, L., 1985. Group selection, altruism, and structured-deme models. Am. Nat. 126, 212–230. Nunney, L., 2000. Altruism, benevolence and culture: commentary discussion of Sober and Wilson’s ‘Unto Others’. J. Consciousness Stud. 7, 231–236. Panchanathan, K., Boyd, R., 2003. A tale of two defectors: the importance of standing for evolution of indirect reciprocity. J. Theor. Biol. 224, 115–126. Pepper, J.W., 2000. Relatedness in trait group models of social evolution. J. Theor. Biol. 206, 355–368. Pepper, J.W., Smuts, B.B., 2002. Assortment through environmental feedback. Am. Nat. 160, 205–213. Price, G.R., 1970. Selection and covariance. Nature 277, 520–521. Queller, D.C., 1985. Kinship, reciprocity and synergism in the evolution of social behavior. Nature 318, 366–367. Queller, D.C., 1992. Quantitative genetics, inclusive fitness, and group selection. Am. Nat. 139, 540–558. Riolo, R.L., Cohen, M.D., Axelrod, R., 2001. Evolution of cooperation without reciprocity. Nature 414, 441–443. Simpson, E.H., 1951. The interpretation of interaction in contingency tables. J. R. Stat. Soc. B 13, 238–241. Sober, E., Wilson, D.S., 1998. Unto Others, the Evolution and Psychology of Unselfish Behavior. Harvard University Press, Cambridge, MA. Sober, E., Wilson, D.S., 2000. Morality and ‘Unto Others’: response to commentary discussions. J. Consciousness Stud. 7, 257–268. Trivers, R.L., 1971. The evolution of reciprocal altruism. Q. Rev. Biol. 46, 35–57. Wade, M.J., 1980. Kin selection: its components. Science 210, 665–667. Wilson, D.S., 1975. A theory of group selection. Proc. Natl Acad. Sci. 72, 143–146. Wilson, D.S., 1979. Structured demes and trait-group variation. Am. Nat. 113, 606–610. Wilson, D.S., 1987. Altruism in Mendelian populations derived from sibling groups: the haystack model revisited. Evolution 41, 1059–1070. Wilson, D.S., 1990. Weak altruism, strong group selection. Oikos 59, 135–140. Wilson, D.S., Colwell, R.K., 1981. Evolution of sex ratio in structured demes. Evolution 35, 882–897.
Journal of Theoretical Biology 228 (2004) 303–313
Strong altruism can evolve in randomly formed groups Jeffrey A. Fletcher*, Martin Zwick Systems Science Ph.D. Program, Portland State University, Portland, OR 97207, USA Received 25 September 2003; accepted 21 January 2004
Abstract Although the conditions under which altruistic behaviors evolve continue to be vigorously debated, there is general agreement that altruistic traits involving an absolute cost to altruists (strong altruism) cannot evolve when populations are structured with randomly formed groups. This conclusion implies that the evolution of such traits depends upon special environmental conditions or additional organismic capabilities that enable altruists to interact with each other more than would be expected with random grouping. Here we show, using both analytic and simulation results, that the positive assortment necessary for strong altruism to evolve does not require these additional mechanisms, but merely that randomly formed groups exist for more than one generation. Conditions favoring the selection of altruists, which are absent when random groups initially form, can naturally arise even after a single generation within groups—and even as the proportion of altruists simultaneously decreases. The gains made by altruists in a second generation within groups can more than compensate for the losses suffered in the first and in this way altruism can ratchet up to high levels. This is true even if altruism is initially rare, migration between groups allowed, homogeneous altruist groups prohibited, population growth restricted, or kin selection precluded. Until now random group formation models have neglected the significance of multigenerational groups—even though such groups are a central feature of classic ‘‘haystack’’ models of the evolution of altruism. We also explore the important role that stochasticity (effectively absent in the original infinite models) plays in the evolution of altruism. The fact that strong altruism can increase when groups are periodically and randomly formed suggests that altruism may evolve more readily and in simpler organisms than is generally appreciated. r 2004 Elsevier Ltd. All rights reserved. Keywords: Altruism; Haystack model; Multilevel selection; Positive assortment; Randomly formed groups
1. Introduction Nearly three decades ago Hamilton (1975) and Wilson (1975) independently developed models which were interpreted as showing that strong altruism (involving an absolute cost to altruists) cannot evolve in randomly formed groups. This conclusion is still generally accepted even among those who debate how best to define altruism and the mechanisms by which it evolves (Hamilton, 1975; Maynard Smith, 1998; Nunney, 1985, 2000; Sober and Wilson, 2000; Wilson, 1975, 1990). Here we challenge this conclusion by exploring what happens when groups exist for more than one generation. Multigenerational groups are a central feature of Maynard Smith’s (1964) classic ‘‘haystack’’ *Corresponding author. Tel.: +1-503-725-4995; fax: +1-503-7258489. E-mail addresses: [email protected] (J.A. Fletcher), [email protected] (M. Zwick). 0022-5193/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2004.01.004
model, but the role of multiple generations within groups was not explored in Hamilton’s (1975) and Wilson’s (1975) models. Although the initial conditions after random group formation favor non-altruists over altruists, paradoxically these conditions can switch to favor altruists after even one generation of selection. Thus even though the overall proportion of altruists decreases after one generation, it can increase even more after a second generation spent within groups. Besides single-generation groups, these original analytic models rely on other simplifying assumptions such as an infinite population and no migration between groups. We begin by showing how strong altruism can evolve under the assumptions of the original models, with the only modification being delayed reformation of random groups. Multigenerational groups introduce additional issues such as interactions among related offspring, persistent homogeneous groups of altruists, and exponential growth of population size. We explore model modifications—preventing altruists from benefiting
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kin, precluding homogeneous groups from forming, and adding a population-level carrying capacity—that mitigate each of these factors. We find that under all these modifications (imposed both separately and concurrently) strong altruism, although dampened, can still evolve in randomly formed multigenerational groups. We then transform the basic analytic model into an evolutionary simulation in which population size is finite and stable. In the simulation model random group formation from a finite population and migration between groups both reduce positive assortment and therefore dampen selection for altruism. On the other hand, increased stochasticity in benefit distribution and culling due to carrying capacity can enhance the likelihood that altruism will evolve compared to processes with minimum stochasticity.
2. Classifications of altruism Both the analytic and simulation models discussed here involve what Pepper (2000) has termed an otheronly altruistic trait because none of the altruist’s benefits come back to itself, as opposed to whole-group traits (also called group-beneficial traits) where the benefit is divided among all group members including the altruist. Wilson (1979, 1990) previously classified altruistic traits in a related but different way as either strong (involving an absolute cost to altruists) or weak (involving only a relative cost to altruists). Other-only altruistic traits are always strong while whole-group traits are strong if the cost to an altruist is greater than its share of the benefit it provides. Note that the same whole-group behavior involving the same sacrifice and provided benefit may be strong or weak depending on group size (Pepper, 2000). In contrast to strong altruism, Wilson (1979, 1990) showed that weakly altruistic traits can increase when groups are randomly formed every generation. That is, for an infinite population where a binomial trait is randomly redistributed every generation, the resulting between-group component of total variance can be enough for weak, but not for strong, altruism to evolve. Nevertheless in finite populations where fitness is relative the distinction between strong and weak altruism may be less important as both types are selected against within groups and require selection (or differential productivity) among groups in order to increase (Wilson, 1979, 1990). In this paper we focus on other-only, strong altruism (the most restrictive situation) to address the random group models of Hamilton and Wilson directly, but the consequences of multigenerational groups and stochasticity also apply to weak, whole-group traits and therefore these traits can even more readily increase via randomly formed groups than was previously shown (Wilson, 1979, 1990).
3. Analytic model We focus on Hamilton’s (1975) model because he developed a formal proof that altruism cannot evolve in single-generation randomly formed groups (Wilson’s (1975) model is similar in all important aspects). In this model a haploid infinite population is randomly subdivided into groups of equal size, n. Group members interact for one generation, affecting each other’s fitness (offspring count), before the population is pooled and then again randomly assigned to new groups. In every generation each altruist behaves in a way that costs itself c offspring and provides a total benefit of b offspring divided evenly among the other n-1 group members. Each non-altruist receives its share of benefits, but does not provide any benefit to others. Therefore, within every group non-altruists have more offspring than altruists, but groups with more altruists have more offspring per capita than groups with less. This is an example of multilevel selection where here selection within groups opposes selection between groups. Hamilton (1975) using Price’s (1970) covariance equation showed that under his model’s assumptions, between-group selection (due to the variance between groups in altruist frequency, p) must always be weaker than average within-group selection (due to the expected variance in the altruistic trait within groups) and therefore the overall frequency of altruists, P, must decrease in every generation. (Capital letters indicate whole population values; small letters indicate group values.) To illustrate this, we calculate D1 P for an infinite binomial distribution, where D1 indicates that the change occurs over one generation within groups, g ¼ 1. The variable g is the number of generations spent within groups before each reformation event. (See Appendix A for model details.) Fig. 1(a) shows how D1 P depends on the level of benefit, b, provided by altruists for different starting P values. (For convenience, all results reported in this paper use c ¼ 1 such that benefit b is also the benefit to cost ratio.) The results shown in Fig. 1(a) are the same for any group size n. Note that as benefit increases, zero is an upper limit on D1 P—hence the conclusion that strong altruism cannot increase under the assumptions of this model for all values of P and n (Hamilton, 1975; Wilson, 1975). Yet quite different results are obtained if groups persist for even one additional generation ðg ¼ 2Þ before random mixing and the formation of new groups. Fig. 1(b) shows how the change in P after two generations within groups, D2 P; depends on benefit values for different starting P values. The only difference between Figs. 1(a) and (b) is that the latter measures the change in altruist frequency after an additional generation spent within groups where the fitness functions are deterministically and recursively
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for an altruistic trait to increase in the next generation is
0.05
rb > c;
∆1P
0 -0.05
P = 0.9 P = 0.75 P = 0.25 P = 0.1
-0.1 -0.15 0
20
40
60
80
100
b
(a) 0.3
∆2P
0.2 0.1 0 P = 0.1 P = 0.25
-0.1 0
20
40
(b)
b
60
P = 0.75 P = 0.9 80
100
0.008 n=2 n=3 n=4 n=5
0.006
∆2P
305
0.004 0.002
where r is the ‘‘coefficient of relatedness’’ or more generally the regression coefficient between the frequency of the trait in recipients and actors (Hamilton, 1972). Thus r is a measure of positive assortment—the degree to which the benefits of altruists fall to other altruists. The value of r differs for other-only and wholegroup traits because in the latter case, but not the former, altruists are recipients of their own actions (Pepper, 2000). We use superscripts w and o, respectively, for whole-group and other-only measures of r. For whole-group traits rw is the between-group variance in p over the total variance in the altruistic trait (Breden, 1990; Frank, 1995a). For an infinite binomial population of randomly formed groups of size n, the variance ratio rw ¼ 1=n: Thus according to Hamilton’s rule (Eq. (1)) the trait increases after one generation if b=n > c; but for whole-group traits this means that an altruist’s share of its benefit must be greater than its cost—this is the definition of weak altruism so as Wilson (1979, 1990) noted only weak traits can increase after one generation. For groups of uniform size the r values are related by the following expression (Pepper, 2000):
0
ro ¼
-0.002 0
20
40
(c)
60
80
100
b
Fig. 1. Change in altruist frequency ðDg PÞ as a function of altruist benefit, b. (a) and (b) compare the effect of different starting P values after one ðg ¼ 1Þ and two ðg ¼ 2Þ generations spent within groups, respectively, where founding group size n ¼ 4 (although in (a) the results are the same for all n); (c) compares the effect of different n for multigenerational groups ðg ¼ 2Þ when altruism is rare—here P ¼ 0:001: The cost c ¼ 1 in all calculations.
applied. In this case strong altruism can clearly increase ðD2 P > 0Þ for sufficient values of benefit. Fig. 1(c) shows that smaller groups give a larger increase in altruist frequency which is consistent with previous findings on the relationship between group size and the evolution of altruistic traits (Avile! s, 1993; Boyd and Richerson, 1988). Additionally Fig. 1(c), for which P = 0.001, shows that strong altruism can increase due to multigenerational groups even when the altruistic trait is rare, although higher benefit levels are needed for D2 P > 0 when P is low.
4. Applying Hamilton’s rule We can also understand these results in terms of Hamilton’s rule (1964) which states that the condition
ð1Þ
nrw 1 : n1
ð2Þ
Therefore ro ¼ 0 for an initial random distribution where rw ¼ 1=n: Obviously there are no positive values of b and c that can satisfy Hamilton’s rule (Eq. (1)) for an other-only (strong) altruistic trait when r ¼ 0 and such traits must decrease. Note however, that any modifications to the model that make ro > 0 can yield an increase in P, given a sufficient value of b. Hamilton (1975) noted that any positive assortment of altruists beyond that produced at random could allow altruism to increase. Surprisingly, for many parameter settings ro increases above zero after one generation of selection— even as the proportion of altruists decreases. That is, this transient one-generation-long ‘‘population viscosity’’ of the original models is enough (without any other mechanisms for creating positive assortment) to create conditions that favor altruism in the following generation. If groups are randomly reformed after this single generation then this gain in positive assortment is destroyed before being used by selection; ro returns to zero and P subsequently declines. On the other hand, additional generations within groups can take advantage of this increased positive assortment so that strong altruism increases, as shown in Fig. 1. (Whether altruism actually increases or not depends on parameters including P, b, and n.) Note that although Hamilton emphasized a ratio of variances in his proof, in this other-only model the
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regression coefficient between actors and recipients, ro, is an easier to interpret measure of the changing conditions affecting altruism as selection occurs. For instance, after one generation of selection (starting with randomly formed groups), the between over total variance, rw, can decrease while ro increases. It is the increase in ro that accurately reflects whether altruism can increase in the next generation. For most parameter settings both measures increase after one generation, but the range of parameters where rw decreases is greater than the range in which ro decreases. In general the r values can decrease when there is a combination of low P, low b, and high n. According to Hamilton’s rule whether altruism increases in the second generation within groups depends on whether ro after the first generation is greater than c/b. Of course, it is not enough for altruism to increase in the second generation for altruism to increase overall. The increases in subsequent generations within groups must make up for any losses in the initial generation(s). Fig. 2(a) shows the expected dynamic change in P values over successive generations when groups persist for one and two generations before
random reformation and Fig. 2(b) shows the concurrent changes in ro. Altruist frequency P decreases monotonically when groups are reformed every generation and ro ¼ 0 before each round of selection. On the other hand when groups exist for two generations, P oscillates (and can ratchet upward). The every-other-generation saw-toothed peaks in P correspond to similar (but offset) oscillating peaks in ro (Fig. 2(b)). Here ro increases after a generation within groups and we indicate the critical c/b value with a solid horizontal line. Troughs on the other hand correspond to global mixing, new group formation, a decrease of ro back to zero and a subsequent decrease in P. In Fig. 2 we also show a case with the same parameters except bigger group size (g ¼ 2; n ¼ 10). Here, although P can increases during the second generations within groups, it is not enough to make up for losses in the first generations. Note that when peaks in ro fail to reach the c/b value (after generation 21 in Fig. 2(b) for g ¼ 2; n ¼ 10), as predicted by Hamilton’s rule, P can no longer increase and instead falls during both generations within groups (Fig. 2(a)).
5. Analytic model modifications 1
g = 1; n = 3 g = 2; n = 10 g = 2; n = 3
P
0.75
0.5
0.25
0 0
5
(a)
10 15 20 total generations
25
30
0.5
ro
0.25 c/b
0
-0.25 0
(b)
5
10 15 20 total generations
25
30
Fig. 2. Calculated dynamics in the analytic model for one and two generations spent within groups ðg ¼ 1; 2Þ for groups of two different sizes (n ¼ 3; n ¼ 10); (a) shows the dynamics in overall altruist frequency P given different starting P values and different sized groups; (b) shows the concurrent change in the regression coefficient between actors and recipients, ro. The critical ro value of c/b is also shown with a solid horizontal line. Here P ¼ 0:1; b ¼ 10; and c ¼ 1: Both ro and P are calculated at the end of the indicated generation and after group reformation if it occurs; (a) and (b) use the same legend.
Multiple generations within groups complicate the simple single-generation model in several ways: (1) kin interactions within groups become possible: (2) the contribution of homogeneous groups of altruists increases—these groups uniquely retain their initial (maximal) level of altruism; (3) the additive frequencydependent fitness functions can now lead to exponential growth of the population. Yet as we demonstrate below, while not inconsequential, none of these factors are essential to explain why strong altruism increases in randomly formed multigenerational groups—especially when altruism is initially rare. The following three paragraphs elaborate on each issue and describe modifications to the basic model to address them. We follow this with a summary of the results produced by each modification. 5.1. No kin selection In the original model groups are formed by randomly selecting individuals from an infinite population and therefore groups contain unrelated individuals. In a second generation within groups, when the benefits provided by an altruist are divided among other group members, some of this benefit (in the form of additional offspring) will fall to those with the same parent as the altruist. In general the proportion of benefit falling to relatives (defined by common ancestry) in subsequent generations will depend on parameters n, b, and P, but this proportion is bounded by 1/n (Appendix B). This
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5.2. No homogeneous groups
0.2
0.1
∆gP
limit is approached for high b and P, but when altruism is initially rare this proportion is much less significant. For example, for groups of size 4, the maximum possible proportion of benefit that could fall to relatives is 1/n=25%, but for P ¼ 0:1 and b ¼ 10 we observe the proportion of benefit given to relatives is actually 0.61% during the second generation within groups. For the same P and b the proportion decreases with larger group size and more generations within groups. To eliminate kin selection we modify our model so that altruists only divide their benefit among non-relatives (Appendix B).
0 g = 2; unmodified g = 2; no homogeneous groups g = 2; no populaiton growth g = 2; no kin benefit g = 2; all 3 modifications g = 1; unmodified
-0.1
-0.2 0
In the infinite population of this model, homogeneous groups of altruists will be randomly created whenever P > 0: These groups are unique in being the only group composition for which p does not decrease with successive generations within groups. They are also the fastest growing groups as they contain no free-riding non-altruists. One might suspect that such homogeneous groups account for altruism being able to increase after multiple generations within groups. To check this we modify our model (Appendix B) such that immediately after group formation all homogeneous groups of altruists have one altruist switched to a non-altruist. Note that this artificially decreases P, making it even harder for altruism to evolve. 5.3. No population growth Even with additive (linear) fitness functions, multiple generations within groups can cause a population to grow exponentially (Wilson, 1987). To study the effect of stable population size we implement a global carrying capacity by scaling the offspring count of all population members each generation by the inverse of the expected overall growth rate during that generation (Appendix B). This holds the population size constant (albeit infinite) at every generation, but allows groups with more altruists to have relatively more offspring each generation than groups with less. 5.4. Modification results Fig. 3 compares the results for each of these modifications with the unmodified model for two generations within groups, g ¼ 2: We also include results for the original model where g ¼ 1: For each of the three modifications D2 P is dampened, but still positive given sufficient benefit. This is true even when all of the modifications are imposed simultaneously. That is, in a model where no benefit is given to kin, homogeneous groups are always corrupted, and population size is held constant, strong altruism can still increase after two generations within groups.
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20
40
60
80
100
b
Fig. 3. Change in altruist frequency ðDg PÞ after two ðg ¼ 2Þ generations within groups as a function of altruist benefit, b, for several modifications of the original binomial model including preventing homogeneous groups from forming, scaling the population size to its original size each generation, and distributing altruist benefit only to the non-relatives of an altruist. The results for the original model after one ðg ¼ 1Þ and two ðg ¼ 2Þ generations are also shown for comparison. The original P ¼ 0:25; n ¼ 4; and c ¼ 1 in all calculations.
In contrast to the unmodified dynamic model shown in Fig. 2 where altruism tends to evolve to P ¼ 1:0 or 0:0 given enough generations, it does not necessarily evolve to saturation under all these modifications. Corrupting homogeneous groups for example necessarily keeps Po1:0: In the case of a population-level carrying capacity, for n ¼ 4; P ¼ 0:1; g ¼ 2; and b ¼ 15; a stable limit cycle is reached in which P oscillates every other generation between 0.616 and 0.636. (Yet as shown in the next section, when stochasticity is introduced populations tend to evolve to one extreme or the other in these models.)
6. Simulation model So far, like Hamilton, we have used the assumption of an infinite population in order to calculate the expected distribution of group compositions when individuals are randomly distributed. But infinity here has two special consequences. First it converts a seemingly stochastic process (random group formation) to a deterministic one—the expected value of ro is produced by every group reformation event. For any finite population, group reformation events will produce ro values that fluctuate both above and below the average value. Second, the expected value of ro is lower (i.e. less than zero) for groups formed randomly from a finite population (compared to an infinite one). This is because in the finite case where ‘‘sampling without replacement’’ is used we have a hypergeometric (rather than binomial) distribution. Once an individual of a certain type is
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assigned to a random group, the probability of assigning another individual of that type is lower than in the first assignment. This results in the formation of less homogeneous groups and more heterogeneous groups (than in the infinite binomial distribution) which decreases the overall assortment—thus on average ro is negative. Based on empirical calculations it appears that the expected negative value of ro depends only on population size (not n or P) and is ro ¼ 1=ðN 1Þ for equal-sized groups randomly formed from a finite population of size N. Particular random group formation events will result in ro values which fluctuate around this lower value. In addition, a finite simulation model can incorporate the consequences of other stochastic processes including the way altruistic benefit is distributed and the way individuals subject to a carrying capacity are eliminated. We now transform the analytic model above into a computer simulation of a finite evolving population and the following features and options (see Appendix C for further details): 1. When reforming groups each individual is assigned at random to an unfilled group (rather than by using the expected distribution). 2. The benefit value used by each group, each generation and the distribution of this benefit to other group members is done in one of two ways: (i) Low stochasticity: the benefit level is the same for all groups during a simulation run and is divided as evenly as possible (in units of whole offspring) with only any remainder distributed randomly among other group members. (ii) High stochasticity: the benefit level used in a group is drawn from a Poisson distribution in whole units where the given benefit value is the mean and each unit of benefit is then distributed at random to other group members. 3. Population size is held constant each generation by a global carrying capacity in one of two ways: (i) Low stochasticity: group sizes are proportionally scaled back (as in the no-growth analytic model), but only whole organisms are removed proportionally with any remainder removed randomly (as below). (ii) High stochasticity: excess population offspring are removed at random (without regard to the altruistic trait or group membership). The first options in 2 and 3 above minimize stochasticity while still preserving the simulation model’s integral organisms, whereas the second options introduce more stochasticity. Fig. 4 compares the change in altruist frequency after 2 generations within
0.2
∆2P
308
0.1
both stochasic options added stochastic elimination added analytic model, no growth stochastic benefit added stochastic group formation
0 0
10
20
30
40
50
b
Fig. 4. Change in altruist frequency after two generations within groups ðD2 PÞ as a function of altruist benefit, b, for several variations of the finite simulation model with no population growth. The base case is stochasticity in random group formation, but otherwise minimal noise. To this case we add high stochasticity in benefit distribution, high stochasticity in implementing carrying capacity, and both options simultaneously. Each data point is the average of 1000 runs done with population size N ¼ 1000: The other parameters for all runs are P ¼ 0:25; n ¼ 4; and c ¼ 1: The results for the infinite analytic model with no population growth using the same parameters are also shown for comparison.
groups, D2 P; for the no-population-growth run of the analytic model (from Fig. 3), the simulation model with stochastic group formation but minimal other stochasticity, and each of the more stochastic choices introduced separately, and then simultaneously. Each data point represents the average of 1000 runs done with different random number seeds. Note that strong altruism evolves less easily in the finite simulation with stochastic group formation (than in the comparable infinite analytic model). From this base, adding more stochasticity in benefit distribution has a slight positive effect whereas adding more stochasticity in elimination due to carrying capacity has a strong positive effect. The latter case results in an even bigger increase in altruism than in the no population growth analytic model. Using both options simultaneously does even better. Our methods of adding stochasticity are somewhat ad hoc and we do not imply that additional stochasticity will necessarily increase D2 P: In fact, the stochasticity in random group formation appears to dampen selection for altruism—D2 P is less than would be expected in an average hypergeometric distribution. The effect of a particular method of introducing random noise will depend on its relative impact on within- and betweengroup selection. We do however show that it is possible for altruism to evolve even more easily in stochastic finite populations than it does in deterministic infinite models. We now explore the long-term behavior of our simulation. Because here there is no mutation, empirically
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% runs P = 1.0
100 80 60
P = 0.90 P = 0.25 P = 0.10 P = 0.05
40 20 0 0
5
(a)
10 b
15
20
10 b
15
20
% runs P = 1.0
100 n=2 n=4 n=6 n=8
80 60 40 20 0 0
(b)
5
Fig. 5. The percentage of simulation runs reaching altruist saturation as a function of altruist benefit, b; (a) compares the effect of different starting P values, where the number of generations spent within groups g ¼ 2 and initial group size n ¼ 4; (b) compares the effect of different n where P ¼ 0:1 and g ¼ 2: The total population size N ¼ 1000 and cost c ¼ 1 in all runs. 1000 runs were done for each unique set of parameters with different random number seeds. All runs were done until P ¼ 1:0 or 0.0. Here the high stochasticity carrying capacity option and the low stochasticity benefit distribution are used.
m = 0.0
m = 0.1
m = 0.2
m = 0.3
100
% runs P = 1.0
we observe that P ¼ 1:0 and 0.0 act as stable equilibrium points and intermediate values do not persist indefinitely. All runs were done until one of these equilibrium points was reached and we use the percentage of 1000 runs reaching altruist saturation, P ¼ 1:0; as a measure of how readily altruism evolves under the given conditions. For all runs and figures described in the rest of this paper we use the high stochasticity option in implementing carrying capacity, but minimum stochasticity in benefit distribution. If the results shown in Fig. 4 apply generally, then these settings are more favorable to the evolution of altruism than the comparable analytic model, but less favorable than if we had used high stochasticity in both processes simultaneously. As was the case for the analytic model, Fig. 5(a) shows that both higher starting P and higher benefit values favor selection for altruism and Fig. 5(b) shows that less altruistic benefit is required to evolve altruism for smaller group sizes. We now investigate the effect of migration in our simulation model where the migration rate, m, specifies the probability that an individual will leave its group during each generation, moving to a randomly selected group (weighted proportionately by group size). The idea here is that larger, thriving groups are proportionately more attractive to migrants, but similar results obtain when migrants join groups at random, independent of group size. Fig. 6 shows how the interaction between the number of generations spent within groups
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80 60 40 20 0 0
2
4
g
6
8
10
Fig. 6. The percentage of simulation runs reaching altruist saturation as a function of the number of generations spent within groups g. Compares the effect of different migration rates, m, where for all runs P ¼ 0:1; g ¼ 2; b ¼ 10; n ¼ 4; N ¼ 1000; and c ¼ 1: 1000 runs were done for each unique set of parameters with different random number seeds. All runs were done until P ¼ 1:0 or 0. For m ¼ 0:4 all runs resulted in P ¼ 0:0 (data not shown). High stochasticity carrying capacity and low stochasticity benefit distribution are used.
and the migration rate influence selection for strong altruism. Predictably, migration lessens selection for altruism (Fig. 6) by working to dampen the positive assortment, ro, between actors and recipients each generation, but for intermediate numbers of generations spent within groups, even at relatively high migration rates (i.e. 30%), strong altruism evolves to saturation in some runs. Fig. 6 also shows that even without migration ðm ¼ 0:0Þ intermediate numbers of generations within groups are most favorable to the evolution of altruism. The advantage of an intermediate number of generations is consistent with similar findings in haystack models (Wilson, 1987) and models of biased sex ratios (Wilson and Colwell, 1981). Note that in the simulations of Fig. 6 it is initially unlikely that any homogeneous groups of altruists will form. With initial P ¼ 0:1 and n ¼ 4 the probability of forming homogeneous altruist groups is one in 10,000 and only 250 groups are formed ðN ¼ 1000; n ¼ 4Þ at each group reformation. Yet, in the absence of homogeneous groups strong altruism can still initially increase overall even as p declines in every group. This is because groups with a higher frequency of altruists grow faster—population P increasing while every group p decreases is an example of Simpson’s paradox (Simpson, 1951; Sober and Wilson, 1998). With more generations within groups P must eventually decrease as the altruists are eliminated from every group. Altruism evolves most readily when the number of generations spent within groups takes full advantage of the increase in P due to Simpson’s paradox, but avoids the inevitable decline in P. Fig. 7 illustrates this tension. Here representative individual runs are shown for 2, 4, and 10 generations within groups using the same parameters as Fig. 6 without migration ðm ¼ 0:0Þ: To aid in comparison the same random number seed (same initial group distribution) is used in all three runs. For 10 generations within groups ðg ¼ 10Þ; reformation clearly takes place well
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310 1 0.8
P
0.6 g=2 g=4 g = 10
0.4 0.2 0 0
5
10 15 20 total generations
25
30
Fig. 7. Dynamics in altruist frequency P for individual simulation runs of g ¼ 2; 4; and 10 generations spent within groups. For all runs initial P ¼ 0:1; b=10, n ¼ 4; N ¼ 1000; and c ¼ 1: To aid in comparison, all three runs were initiated with the same random number seed (same initial distribution into groups). High stochasticity carrying capacity and low stochasticity benefit distribution are used.
after peak P values are reached and altruism eventually goes extinct. For g ¼ 4; reformation takes place near peak P values and altruism rapidly evolves towards saturation—even though altruism always decreases in the first generation after reformation. (This is true for g ¼ 10 also, but harder to see as the rate of decline after reformation matches the rate before.) On the other hand when groups are reformed every other generation ðg ¼ 2Þ; the potential additional increase in P that would result from staying within groups longer is lost and altruism increases more gradually. Note that the initial increase in P in these three runs takes place in the absence of homogeneous groups. For g ¼ 2 and 4 no such groups are formed until P reaches about 0.3 (initial P ¼ 0:1) and in the case of g ¼ 10 homogeneous groups never formed. In contrast, in the infinite analytic model homogeneous groups are always initially present and more generations within groups can allow these fastestgrowing groups to become more and more dominant, even if initially rare.
7. Conclusion The main purpose of this paper is to demonstrate that strong altruism can evolve in randomly formed groups and thereby challenge a presumed theoretic limitation on the evolution of altruism. Although allowing groups to last more than one generation introduces new complications, we have demonstrated that kin selection, homogeneous groups, and population expansion are not essential to account for this phenomenon. The fundamental explanation is that, for many initial conditions, after even just one generation of selection in randomly formed groups, the positive assortment between altruists and their potential recipients increases (above the expected initial value for randomly formed groups) as measured by the regression coefficient, ro. The groups
that are by chance initially dominated by altruists grow larger compared to other groups and even though the fraction of altruists declines in these groups, the absolute number of altruists poised to benefit other altruists in a subsequent generation increases. On the other hand, the groups that are by chance dominated by non-altruists do not grow as large and the relatively few altruists in these groups are eliminated or greatly diminished after one to several generations within groups. This also increases positive assortment as these non-altruists are stuck with each other and will receive less benefit from altruists than they did in the first generation. Of course the few non-altruists lucky enough to end up in altruistdominated groups are the fittest individuals, but overall the conditions that favored non-altruists in the initial random distribution can switch to favor altruists in subsequent generations. We emphasize again that even when groups are multigenerational, the vast majority of the benefit provided by altruists will fall to non-relatives—especially when altruism is initially rare (Appendix B). Altruism evolves due to the positive assortment among heritable helping behaviors regardless of whether there is a positive assortment among relatives by descent. The regression coefficient used here, ro, measures the former. This positive assortment can be viewed equivalently (Frank, 1998; Queller, 1985, 1992; Sober and Wilson, 1998; Wade, 1980) as causing selection on the altruistic trait (allele) via inclusive fitness or as causing selection among groups that vary in their trait composition. While interactions among kin in nature no doubt often contribute to the positive assortment of altruistic traits, kin interactions are not in themselves a requirement for altruism to evolve. Whether strong altruism evolves in nature via mechanisms similar to those illustrated here will depend on the degree to which the assumptions of these models are representative of natural conditions. For instance, in both the analytic and simulation models we demonstrated that strong altruism can evolve even when population size is held constant by a global carrying capacity. In nature, in addition to population-level limits on growth there are often limits on group size. While not explored here, group-level limits will dampen between-group selection for altruism, so further investigation is needed to elucidate the relative import of global vs. local levels of population control in the evolution of altruism. A lack of mutation is also unrealistic. We experimented with mutation in our models (data not shown), but in the simple binary genetics used here a mutation that switches behavioral types exerts pressure towards P ¼ 0:5 and thus favors altruism when P is initially low. This is because the more common type experiences more mutations. Even if this bias could be compensated for, low mutation rates are unlikely to alter our basic results, which are robust
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under fairly high levels of migration among groups and even when homogeneous groups of altruists are ‘‘mutated’’ to contain at least one non-altruist. This model started with the original assumptions of the random group models (Hamilton, 1975; Wilson, 1975) and added the idea of multigenerational groups from haystack models (Maynard Smith, 1964; Wilson, 1987). Just as Wilson (1987) created a simulation model to study a more realistic version of Maynard Smith’s (1964) original haystack model, we have created a simulation model that adds finite population size, stochasticity, and mutigenerational groups to the original analytic random group models. Whereas Wilson’s (1987) haystack simulation corrected the ‘‘worst case’’ assumption made by Maynard Smith (1964) that groups would persist until altruism was eliminated in all mixed groups; here we correct an opposite ‘‘worst case’’ assumption made in random group models that groups only exist for a single generation. As demonstrated here and in the haystack simulations (Wilson, 1987), an intermediate number of generations within groups is most favorable to the evolution of altruism. Maynard Smith (1998) in discussing different views on the evolution of altruism recently echoed the original findings of Hamilton (1975) and Wilson (1975) and the current consensus opinion when he wrote: ‘‘If costs and benefits combine additively, and groups are formed randomly, then altruism cannot evolve. But if altruists tend to associate with altruists, and non-altruists with non-altruists, then altruism can evolve. This conclusion is agreed.’’ Many mechanisms which result in a positive assortment among self-sacrificing behaviors have been proposed including passive methods such as foraging in non-uniform resource distributions which can be depleted (Pepper and Smuts, 2002), continuous population viscosity with periodic environmental disturbances (Mitteldorf and Wilson, 2000), the coevolution of group joining and cooperative behaviors (Avile! s, 2002), and the presence of non-participants (Hauert et al., 2002), as well as active methods such as kin recognition (Gamboa et al., 1991), conditional strategies based on past actions (Axelrod and Hamilton, 1981; Trivers, 1971) or reputation (Nowak and Sigmund, 1998; Panchanathan and Boyd, 2003), policing (Frank, 1995b, 2003), punishment of non-altruists (Boyd et al., 2003; Boyd and Richerson, 1992; Fehr and G.achter, 2002), the coevolution of cultural institutions that constrain individual behaviors (Bowles et al., 2003), and even recognition of arbitrary tags (Riolo et al., 2001). Here we have shown in both deterministic and stochastic models that when groups exist for more than one generation such specific or more complex mechanisms for creating positive assortment, although certainly important if present, are not needed—the positive assortment that develops between randomly created multigenerational groups can suffice
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for between-group selection to dominate within-group selection and thus for strong altruism to evolve.
Acknowledgements We wish to thank Leticia Avile! s and Mark Bedau for helpful comments on an earlier draft of this manuscript and two anonymous reviewers for detailed and useful suggestions. We also thank Gene Enneking, Len Nunney, and Tad Shannon for helpful discussions.
Appendix A. Analytic model Here we describe Hamilton’s original model with recursion added to accommodate multiple generations within groups. If ag, sg, and ng are, respectively, the number of altruists, non-altruists (selfish individuals), and total individuals in a group after g generations spent within groups, then: ag1 1 ag ¼ ag1 1 þ b c ; ðA:1Þ ng1 1 ag1 sg ¼ sg1 1 þ b and ðA:2Þ ng1 1 ag1 ng ¼ ng1 1 þ ðb cÞ : ðA:3Þ ng1 In Hamilton’s model g is always one, but in our model we vary g by using these equations recursively— inputting the results from one generation into the calculations for the next. Note that when first formed all groups are size n, but after reproduction group sizes vary. (Terms without g subscripts indicate initial values, i.e. n is n0.) The overall number of altruists, Ag ; and individuals, Ng ; in the population after g generation within groups is then the number contributed (after g generations) by groups of every possible original composition (a = 0 to n) times the number of such groups expected in a random binomial distribution. If G is the total number of groups, then the expected count of groups with a initial altruists out of n group members is ! n hðaÞ ¼ G Pa ð1 PÞðnaÞ : ðA:4Þ a The total population values after g generations spent within groups are then given by n X Ag ¼ hðiÞ ag ðiÞ and ðA:5Þ i¼0
Ng ¼
n X i¼0
hðiÞ ng ðiÞ;
ðA:6Þ
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where ag(i) is the ag value from Eq. (A.1) given the particular starting a value specified by the index i and similarly for ng(i). Although G is infinite, it cancels in the calculation of Pg ¼ Ag =Ng and Dg P ¼ Pg 2P.
Appendix B. Analytic model modifications Here we describe modifications to the analytic model such that altruists do not give benefit to kin, homogeneous groups are disallowed, and the population size is kept constant.
B.2. No homogeneous groups In an infinite population there will always be some homogeneous groups whenever P > 0: For this modification, each time groups are randomly formed we eliminate homogeneous groups by simply converting all groups where the altruist number a ¼ n to groups where a ¼ n21: In this way all homogeneous groups of altruists have one member switched to a non-altruist. Because non-altruists always increase faster than altruists, groups become more non-homogeneous with each successive generation. Note that this modification causes P to decrease, but this effect is small when P is small.
B.1. No benefit to kin The modified fitness functions for when altruists only give to non-relatives are implemented by substituting the size of altruist kin groups, k, for the minus-one term in Eqs. (A.1) and (A.2). The minus-one term subtracted the altruist from the number of its beneficiaries; here we subtract the altruist’s kin (those having a common ancestor) as well. A preceding superscript k is used to designate fitness calculations that subtract k instead of one from a group’s altruist count and group size: k ag1 kg1 k k ag ¼ ag1 1 þ b c ; ðB:1Þ ng1 kg1 k
sg ¼ k sg1 1 þ b
ag1 ; ng1 kg1 k
ðB:2Þ
where the size of a kin group of altruists in generation g is given by ag kg ¼ kg1 : ðB:3Þ ag1 This is the size of the kin group in the last generation times an altruist’s clutch size for this generation. The initial k value k0 ¼ 1 (altruists are only related to themselves). Shifting benefit from kin to non-kin in this way does not affect the total group size and Eq. (A.3) works for calculating ng. Note that in the unmodified model the average proportion of a group that is related to an altruist, kg/ng, can never be above 1/n and therefore the proportion of an altruist’s benefit that falls to kin (kg–1)/(ng–1) is also bounded by 1/n. To see this note that kg/ng will be largest within homogeneous groups of altruists compared to mixed groups. In such groups (given our convention that c ¼ 1) kg is multiplied by b each generation and total group size also increases with b. Therefore the proportion kg/ng remains at its original value of 1/n. In all other groups this proportion falls with successive generations. Only when P is high (so that homogeneous altruist groups are common) or when b is high (so homogeneous altruist groups grow proportionally bigger than other groups) is this limit approached.
B.3. No population growth The global carrying capacity is implemented by scaling back all offspring numbers each generation by Ng1 =Ng where Ng is first calculated without scaling. We use a preceding asterisk to denote values calculated with scaling. For instance, the number of altruists in a group after g generations with scaling is, a ¼ a Ng1 g g Ng
ðB:4Þ
and similarly for group size ng, where scaling is imposed at each recursion (generation). Whole population values with scaling Ag and Ng then sum over ag and ng instead of ag and ng, respectively, in Eqs. (A.5) and (A.6) and Pg=Ag/Ng.
Appendix C. Computer simulation model For each run of the model, individuals ðN ¼ 1000Þ are initially randomly distributed into groups of size n using a random number generator to assign individuals to unfilled groups. The proportion of altruists and nonaltruists is determined by the starting P value. The sequential steps of the simulation are then: 1. In each group the new number of altruists and nonaltruists (to the closest whole individual) are determined (using either the low or high stochasticity method described in the text). 2. Individuals are eliminated (using either the low or high stochasticity method described in the text) until the original population size N is reached. 3. If g generations have passed within groups since the last group reformation, all individuals are randomly assigned to new groups of size n; otherwise if the migration rate, m, is greater than zero, mN individuals are chosen at random from the whole population and moved to new random locations in the population array which is ordered by
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groups—consequently larger groups are proportionately more likely to send out and receive migrants. These steps are repeated until an equilibrium at P ¼ 0:0 or 1.0 is reached. For runs where n ¼ 6; N was 1002 instead of 1000 and initial P ¼ 0:0998 instead of 0.1000 to allow an even distribution into groups.
References Avil!es, L., 1993. Interdemic selection and the sex ratio: a social spider perspective. Am. Nat. 142, 320–345. Avil!es, L., 2002. Solving the freeloaders paradox: genetic associations and frequency-dependent selection in the evolution of cooperation among nonrelatives. Proc. Natl Acad. Sci. 99, 14268–14273. Axelrod, R., Hamilton, W.D., 1981. The evolution of cooperation. Science 211, 1390–1396. Bowles, S., Choi, J.-K., Hopfensitz, A., 2003. The co-evolution of individual behaviors and social institutions. J. Theor. Biol. 223, 135–147. Boyd, R., Richerson, P.J., 1988. The evolution of reciprocity in sizeable groups. J. Theor. Biol. 132, 337–356. Boyd, R., Richerson, P.J., 1992. Punishment allows the evolution of cooperation (or anything else) in sizable groups. Ethol. Sociobiol. 13, 171–195. Boyd, R., Gintis, H., Bowles, S., Richerson, P.J., 2003. The evolution of altruistic punishment. Proc. Natl Acad. Sci. 100, 3531–3535. Breden, F., 1990. Partitioning of covariance as a method for studying kin selection. Trends Ecol. Evol. 5, 224–228. Fehr, E., G.achter, S., 2002. Altruistic punishment in humans. Nature 415, 137–140. Frank, S.A., 1995a. George price’s contributions to evolutionary genetics. J. Theor. Biol. 175, 373–388. Frank, S.A., 1995b. Mutual policing and repression of competition in the evolution of cooperative groups. Nature 377, 520–522. Frank, S.A., 1998. Foundations of Social Evolution. Princeton University Press, Princeton. Frank, S.A., 2003. Perspective: repression of competition and the evolution of cooperation. Evolution 57, 693–705. Gamboa, G.J., Reeve, H.K., Holmes, W.G., 1991. Conceptual issues and methodology in kin-recognition research: a critical discussion. Ethology 88, 109–127. Hamilton, W.D., 1964. The genetical evolution of social behavior i and ii. J. Theor. Biol. 7, 1–52. Hamilton, W.D., 1972. Altruism and related phenomena, mainly in social insects. Annu. Rev. Ecol. Syst. 3, 193–232. Hamilton, W.D., 1975. Innate social aptitudes of man: an approach from evolutionary genetics. In: Fox, R. (Ed.), Biosocial Anthropology. Wiley, New York, pp. 133–155.
313
Hauert, C., De Monte, S., Hofbauer, J., Sigmund, K., 2002. Volunteering as red queen mechanism for cooperators in public goods games. Science 296, 1129–1132. Maynard Smith, J., 1964. Groups selection and kin selection. Nature 201, 1145–1147. Maynard Smith, J., 1998. The origin of altruism. Nature 393, 639–640. Mitteldorf, J., Wilson, D.S., 2000. Population viscosity and the evolution of altruism. J. Theor. Biol. 204, 481–496. Nowak, M.A., Sigmund, K., 1998. Evolution of indirect reciprocity by image scoring. Nature 393, 573–577. Nunney, L., 1985. Group selection, altruism, and structured-deme models. Am. Nat. 126, 212–230. Nunney, L., 2000. Altruism, benevolence and culture: commentary discussion of Sober and Wilson’s ‘Unto Others’. J. Consciousness Stud. 7, 231–236. Panchanathan, K., Boyd, R., 2003. A tale of two defectors: the importance of standing for evolution of indirect reciprocity. J. Theor. Biol. 224, 115–126. Pepper, J.W., 2000. Relatedness in trait group models of social evolution. J. Theor. Biol. 206, 355–368. Pepper, J.W., Smuts, B.B., 2002. Assortment through environmental feedback. Am. Nat. 160, 205–213. Price, G.R., 1970. Selection and covariance. Nature 277, 520–521. Queller, D.C., 1985. Kinship, reciprocity and synergism in the evolution of social behavior. Nature 318, 366–367. Queller, D.C., 1992. Quantitative genetics, inclusive fitness, and group selection. Am. Nat. 139, 540–558. Riolo, R.L., Cohen, M.D., Axelrod, R., 2001. Evolution of cooperation without reciprocity. Nature 414, 441–443. Simpson, E.H., 1951. The interpretation of interaction in contingency tables. J. R. Stat. Soc. B 13, 238–241. Sober, E., Wilson, D.S., 1998. Unto Others, the Evolution and Psychology of Unselfish Behavior. Harvard University Press, Cambridge, MA. Sober, E., Wilson, D.S., 2000. Morality and ‘Unto Others’: response to commentary discussions. J. Consciousness Stud. 7, 257–268. Trivers, R.L., 1971. The evolution of reciprocal altruism. Q. Rev. Biol. 46, 35–57. Wade, M.J., 1980. Kin selection: its components. Science 210, 665–667. Wilson, D.S., 1975. A theory of group selection. Proc. Natl Acad. Sci. 72, 143–146. Wilson, D.S., 1979. Structured demes and trait-group variation. Am. Nat. 113, 606–610. Wilson, D.S., 1987. Altruism in Mendelian populations derived from sibling groups: the haystack model revisited. Evolution 41, 1059–1070. Wilson, D.S., 1990. Weak altruism, strong group selection. Oikos 59, 135–140. Wilson, D.S., Colwell, R.K., 1981. Evolution of sex ratio in structured demes. Evolution 35, 882–897.
